* Corresponding author. Chemical Engineering Science 56 (2001) 1317}1325 Determination of kinetic expressions from the frequency response of a catalytic reactor * theoretical and experimental investigations Abdul R. Garayhi, Frerich J. Keil* Technical University of Hamburg-Harburg, Department of Chemical Engineering, Eissendorfer Str. 38, D-21071 Hamburg, Germany Abstract A new technique is presented for the investigation of non-stationary reaction systems. Reliable quantitative information is obtained forthedescriptionofnon-stationaryoperation.Adetaileddescriptionofthereactionmechanismispossibleanddi!erent mechanisms with the same steady-state behavior can be discriminated. Two models for the evaluation are presented, an analytical expression for a fast parameter determination and a rigorous approach for the determination of the elementary kinetic expressions. The new technique is used for the investigation of the total combustion of diluted methane on a Pd-support catalyst. A Lan- gmuir}Hinshelwood mechanism was found, a Eley}Rideal mechanism could be excluded. The detailed chemical mechanism is presented and kinetic constants for the elementary steps are determined.  2001 Elsevier Science Ltd. All rights reserved. 1. Introduction The non-steady-state operation of chemical plants is frequently employed. One example is classical batch op- eration, but new reactor concepts like the reverse-#ow reactor are also used. In these cases, the non-stationary concept is used to realize a more economic operation than can be obtained in a quasi-stationary operation. Other non-stationary processes which were analyzed in the past are start-up and shut-down operations and shifts from one steady state to another as well as fault situ- ations. Often, reaction rate expressions that were deter- mined under steady-state conditions are not suitable for the description of non-stationary operation, as could be demonstrated in many examples. Qualitative non-sta- tionary experiments to "nd indications of the catalytic mechanism which employ step-response experiments have been in use for the investigation of catalytic systems for several years. Usually, a concentration step is applied at the inlet of a laboratory reactor, the characteristic response may indicate the rate-determining step of the catalytic reaction or may show structural changes in the active part of the catalyst (Kobayashi & Kobayashi, 1974). For a quantitative description, these step-response experiments are only suitable in a limited number of cases, because the transient period of the system is quite short (Garayhi & Keil, 1999). Furthermore, it is some- times desirable to obtain information on the inherent non-stationary behavior, while the step response is dom- inated by well-known initial e!ects like mass-transport limitation or ignition and extinction of the reactor, which may not occur during the following non-stationary op- eration. To avoid these disadvantages, the method pre- sented here was developed. A consistant quantitative description of the dynamic behavior is determined from the frequency response which is well established in other "elds of engineering like control engineering. For linear systems, the output signal to a sinusoidal input signal is again a sine wave, which usually has a di!erent ampli- tude and is dephased to the input. The frequency re- sponse relates the amplitude ratio of output and input signal and the phase shift to the exciting frequency (Luyben, 1990). For non-linear systems, the output signal to a sinusoidal input is no longer a pure sine wave, however, the output signal is usually periodical. Often, the period of the output is the same as the period of the sine input signal, although sometimes the periods may be di!erent, e.g. period doubling may occur (Kuznetsov, 1998). If the period is conserved, the best approximating sine signal is found from the zeroth and "rst Fourier coe$cients. This sine signal is used to construct the frequency response of the non-linear system. Application of frequency response measurements in the "eldofChemicalEngineeringhavebeeninuseforthe 0009-2509/01/$-see front matter  2001 Elsevier Science Ltd. All rights reserved. PII:S0009-2509(00)00354-7